(a) Suppose that (REF ) holds. As observed in [1]} (proof of Theorem 2), it follows from the Berry–Esseen theorem that
\(\frac{{\mathbb {E}}\,f_k(C_N)}{\binom{N}{k}}&=\frac{\Phi \left(\frac{2d-N-k-1}{\sqrt{N-k-1}}\right)+O\left(\frac{1}{\sqrt{N-k-1}}\right)}{\Phi \left(\frac{2d-N-1}{\sqrt{N-1}}\right)+O\left(\frac{1}{\sqrt{N-1}}\right)}.\) 
